Most economical way to backup when my USB ports are dead?

It might be slow but you could end up transferring your data over a shared user account and your home network. I have two Macs and I have a shared folder on one that I use for grabbing data off that one over my Ethernet connection.

Unfortunately "cheap" usually also requires skill sets or being able to develop them. We spend minimal money on maintaining things in our house because when something goes wrong with the plumbing I learn how to be a plumber; when something goes wrong with the computer I learn how to be an IT specialist; when something goes wrong with the car I am out in the garage with a car manual and my toolbox, etc. I've not worked on your exact model computer but when my MacBook died it took two screwdrivers and about 10 minutes to get my old drive out of it. 5 of those minutes were reading on the Internet how to do it. ;-)

When are you getting this new computer? If soon, you could buy a $25-$30 dongle that lets you attach a bare drive to a computer (I bought one from macsales.com several years ago), then remove the drive from the old computer and attach it to the new one. Or if you want to continue to use the old drive (bear in mind that given its age it may not last too much longer anyway) you could pay for a proper enclosure. In fact my inclination would be to go for a Voyager docking station. I have two of those with multiple bare drives I can swap out for various purposes.

Go to macsales.com and look at their drive replacement video for a 2012 MBP. The drive is really simple to remove. You can then use a USB-C to SATA cable to connect the drive to the new computer. Or buy an enclosure (most require that you just slide the drive in and snap the enclosure shut) and use it as extra storage.

If you only have a couple gigabyte of data to transfer you could use a free cloud account (Google, DropBox) etc but that's kinda slow.

You don't have to leave it installed in that external enclosure. When you're finished using it to set up a replacement MBP, just reinstall it in the original one.

It's not particularly complicated, but if you contact OWC / MacSales as dwb suggested they can talk you through it, or even (for a fee) do it for you.

Most economical, as in cheapest?

When you obtain a replacement MacBook Pro, physically remove the internal hard disk drive and install it in an external USB enclosure.

Total cost < $10 US.